v2007.09.13 - Convex Optimization
v2007.09.13 - Convex Optimization v2007.09.13 - Convex Optimization
58 CHAPTER 2. CONVEX GEOMETRYsvec ∂ S 2 +I[ α ββ γ]γα√2βFigure 14: Two Fantopes. Circle, (radius 1 √2) shown here on boundary ofpositive semidefinite cone S 2 + in isometrically isomorphic R 3 from Figure 31,comprises boundary of a Fantope (79) in this dimension (k = 1, N = 2).Lone point illustrated is identity matrix I and that Fantope correspondingto k = 2, N = 2. (View is from inside PSD cone looking toward origin.)
2.4. HALFSPACE, HYPERPLANE 59of dimension k(N −√k(1 k + 1) and radius − k ) centered at kI along2 2 N Nthe ray (base 0) through the identity matrix I in isomorphic vector spaceR N(N+1)/2 (2.2.2.1).Figure 14 illustrates extreme points (81) comprising the boundary of aFantope, the boundary of a disc corresponding to k = 1, N = 2 ; but thatcircumscription does not hold in higher dimension. (2.9.2.5) 2.3.3 Conic hullIn terms of a finite-length point list (or set) arranged columnar in X ∈ R n×N(65), its conic hull is expressedK ∆ = cone {x l , l=1... N} = coneX = {Xa | a ≽ 0} ⊆ R n (83)id est, every nonnegative combination of points from the list. The conic hullof any finite-length list forms a polyhedral cone [147,A.4.3] (2.12.1.0.1;e.g., Figure 15); the smallest closed convex cone that contains the list.By convention, the aberration [245,2.1]Given some arbitrary set C , it is apparent2.3.4 Vertex-descriptioncone ∅ ∆ = {0} (84)conv C ⊆ cone C (85)The conditions in (67), (75), and (83) respectively define an affinecombination, convex combination, and conic combination of elements fromthe set or list. Whenever a Euclidean body can be described as somehull or span of a set of points, then that representation is loosely calleda vertex-description.2.4 Halfspace, HyperplaneA two-dimensional affine subset is called a plane. An (n −1)-dimensionalaffine subset in R n is called a hyperplane. [228] [147] Every hyperplanepartially bounds a halfspace (which is convex but not affine).
- Page 7 and 8: PreludeThe constant demands of my d
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- Page 13 and 14: List of Figures1 Overview 191 Orion
- Page 15 and 16: LIST OF FIGURES 1559 Quadratic func
- Page 17 and 18: LIST OF FIGURES 17E Projection 5791
- Page 19 and 20: Chapter 1OverviewConvex Optimizatio
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- Page 23 and 24: 23Figure 4: This coarsely discretiz
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- Page 29 and 30: its membership to the EDM cone. The
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- Page 33 and 34: Chapter 2Convex geometryConvexity h
- Page 35 and 36: 2.1. CONVEX SET 35Figure 9: A slab
- Page 37 and 38: 2.1. CONVEX SET 372.1.6 empty set v
- Page 39 and 40: 2.1. CONVEX SET 392.1.7.1 Line inte
- Page 41 and 42: 2.1. CONVEX SET 41(a)R 2(b)R 3(c)(d
- Page 43 and 44: 2.1. CONVEX SET 43This theorem in c
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- Page 53 and 54: 2.3. HULLS 53Figure 12: Convex hull
- Page 55 and 56: 2.3. HULLS 55Aaffine hull (drawn tr
- Page 57: 2.3. HULLS 57The union of relative
- Page 61 and 62: 2.4. HALFSPACE, HYPERPLANE 61H +ay
- Page 63 and 64: 2.4. HALFSPACE, HYPERPLANE 63Inters
- Page 65 and 66: 2.4. HALFSPACE, HYPERPLANE 65Conver
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- Page 69 and 70: 2.4. HALFSPACE, HYPERPLANE 69tradit
- Page 71 and 72: 2.4. HALFSPACE, HYPERPLANE 71There
- Page 73 and 74: 2.5. SUBSPACE REPRESENTATIONS 732.5
- Page 75 and 76: 2.5. SUBSPACE REPRESENTATIONS 752.5
- Page 77 and 78: 2.6. EXTREME, EXPOSED 77In other wo
- Page 79 and 80: 2.6. EXTREME, EXPOSED 792.6.1 Expos
- Page 81 and 82: 2.7. CONES 812.6.1.3.1 Definition.
- Page 83 and 84: 2.7. CONES 830Figure 24: Boundary o
- Page 85 and 86: 2.7. CONES 852.7.2 Convex coneWe ca
- Page 87 and 88: 2.7. CONES 87Thus the simplest and
- Page 89 and 90: 2.7. CONES 89nomenclature generaliz
- Page 91 and 92: 2.8. CONE BOUNDARY 91Proper cone {0
- Page 93 and 94: 2.8. CONE BOUNDARY 93the same extre
- Page 96 and 97: 96 CHAPTER 2. CONVEX GEOMETRYBCADFi
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58 CHAPTER 2. CONVEX GEOMETRYsvec ∂ S 2 +I[ α ββ γ]γα√2βFigure 14: Two Fantopes. Circle, (radius 1 √2) shown here on boundary ofpositive semidefinite cone S 2 + in isometrically isomorphic R 3 from Figure 31,comprises boundary of a Fantope (79) in this dimension (k = 1, N = 2).Lone point illustrated is identity matrix I and that Fantope correspondingto k = 2, N = 2. (View is from inside PSD cone looking toward origin.)
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